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A238208 The total number of 1's in all partitions of n into an odd number of distinct parts. 1
0, 1, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 24, 28, 33, 38, 45, 52, 60, 69, 80, 91, 105, 120, 137, 156, 178, 202, 230, 261, 295, 334, 378, 426, 481, 542, 609, 685, 769, 862 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

The g.f. for "number of k's" is (1/2)*x^k/(1+x^k)*prod(n>=1, 1+x^n)+(1/2)*x^k/(1-x^k)*prod(n>=1, 1-x^n).

Or: the number of partitions of n-1 into an even number of distinct parts >=2. - R. J. Mathar, May 11 2016

LINKS

R. J. Mathar, Table of n, a(n) for n = 0..60

FORMULA

a(n)=sum_{j=1..round(n/2)}A067661(n-(2*j-1))-sum_{j=1..floor(n/2))}A067659(n-2*j).

G.f.: (1/2)*x/(1+x)*prod(n>=1,1+x^n)+(1/2)*x/(1-x)*prod(n>=1,1-x^n).

EXAMPLE

a(10)=3 because the partitions in question are: 7+2+1, 6+3+1, 5+4+1.

MAPLE

A238208 := proc(n)

    local a, L, Lset;

    a := 0 ;

    L := combinat[firstpart](n) ;

    while true do

        # check that parts are distinct

        Lset := convert(L, set) ;

        if nops(L) = nops(Lset) then

            # check that number is odd

            if type(nops(L), 'odd') then

                if 1 in Lset then

                    a := a+1 ;

                end if;

            end if;

        end if;

        L := combinat[nextpart](L) ;

        if L = FAIL then

            return a;

        end if;

    end do:

    a ;

end proc: # R. J. Mathar, May 11 2016

CROSSREFS

Cf. A067659, A067661.

Sequence in context: A185327 A210717 A171962 * A029028 A240572 A029072

Adjacent sequences:  A238205 A238206 A238207 * A238209 A238210 A238211

KEYWORD

nonn

AUTHOR

Mircea Merca, Feb 20 2014

STATUS

approved

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Last modified December 13 09:05 EST 2017. Contains 295957 sequences.